To learn algebraic geometry by yourself you could use some of the first books I recommend in my Amazon list: amazon.com/lm/RHQS8Y3V7LJRQ/ref=cm_pdp_lm_title_1 Above all, you should try in this order Beltrametti et al., Hulek, Shafarevich and Perrin. You can also search the web and download the pdf courses by Dolgachev, Gathmann, Milne and Fulton (and Debarre in French or Manetti in Italian)
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Javier ÁlvarezFeb 20 '11 at 1:32

1 Answer
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It suffices to say where $x$ and $y$ go because they generate $K[x, y]$ (by the universal property).

The universal property tells you that a $K$-algebra homomorphism out of a polynomial ring over $K$ is determined by the images of the indeterminates.

The targets of $x$ and $y$ are chosen so that the ideal of the homomorphism is $I(Y)$. A different way of seeing this is to note that there is a (surjective) morphism $\mathbb{A}^1 \to Y$ sending $t \mapsto (t, t^2)$ — a morphism of affine varieties corresponds to a $K$-algebra homomorphism of the coordinate rings (but in the opposite direction!) and vice-versa.